Procesado de Señales e Imágenes Médicas

Ingeniería Biomédica

Ph.D. Pablo Eduardo Caicedo Rodríguez

2024-08-12

Introduction to Wavelet Transform

Introduction

  • It’s a mathematical tool for signal decomposition, like Fourier’s Transform.
  • Just as the Fourier transform decomposes a signal into a series of sine and cosine functions, the wavelet transform does so using a set of functions known as wavelets.
  • Wavelets are functions generated by scaling and shifting a base function known as the mother wavelet.

Introduction

Introduction

  • Morlet: Popular for time-frequency analysis in EEG and ECG.
  • Mexican Hat (Ricker): Often used in spike detection in neural signals.
  • Haar: Useful in quick decomposition of signals and feature extraction.
  • Daubechies: Frequently used in ECG signal denoising and compression.
  • Symlet: Another option for signal processing and feature extraction in EEG.
  • Coiflet: Useful for denoising and baseline correction in biomedical signals.

Introduction

  • Have a mean of zero (to capture details in the signal).
  • Be square integrable (finite energy).
  • Satisfy the admissibility condition on its Fourier transform.
  • Be oscillatory to capture frequency information.
  • (Optionally) have compact support for efficient computation and localization.

The function must have an average value of zero. Mathematically, this is expressed as:

\[\int_{-\infty}^{\infty} \psi(t) \, dt = 0\]

This condition ensures that the wavelet can detect changes or “details” in the signal rather than its average or constant components.

The function \(\psi(t)\) must be square integrable, meaning it has finite energy:

\[\int_{-\infty}^{\infty} |\psi(t)|^2 \, dt < \infty\]

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